Cohen-stable families of subsets of integers

A maximal almost disjoint (mad) family A subset of or equal to [omega](omeg a) is Cohen-stable if and only if it remains maximal in any Cohen generic e xtension. Otherwise it is Cohen-unstable. It is shown that a mad family, A, is Cohen-unstable if and only if there is a bijection G from omega to the rationals such that the sets G[A]. A is an element of A are nowhere dense. An N-0-mad family, A, is a mad family with the property that given any coun table family B subset of C [omega](omega) such that each element of B meets infinitely many elements of A in an infinite set there is an clement of A meeting each element of B in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist N-0-mad families. Either of the conditions b = c or a < cov(K) implies that there exist Cohen-stable ma d families. Similar results are obtained for splitting families. For exampl e, a splitting family, S, is Cohen-unstable if and only if there is a bijec tion G from <omega> to the rationals such that the boundaries of the sets G [S]. S is an element of S are nowhere dense. Also, Cohen-stable splitting f amilies of cardinality less than or equal to kappa exist if and only if N-0 -splitting families of cardinality less than or equal to kappa exist.